Why isn't conditional probability sufficient to describe causality?
Perhaps the shortest answer to this question is that Bayes' Theorem itself allows us to easily change the direction of a conditional probability:

$$
P(A|B) = \frac{P(B|A)P(A)}{P(B)}
$$

So if you have $P(B|A)$, $P(A)$, and $P(B)$, we can determine $P(A|B)$, and similarly you can determine $P(B|A)$ from $P(A|B)$, $P(B)$ and $P(A)$. Just by looking at $P(B|A)$ and $P(A|B)$, it is therefore impossible to tell what the causal direction is (if any).

In fact, probabilistic inference usually works the other way round: When there is a known causal relation, say from diseases $A$ to symptoms $B$, we usually have $P(B|A)$, and are interested in the diagnostic reasoning task of determining $P(A|B)$ from that. (The only other thing we need for that is the prior probability $P(A)$ since $P(B)$ is just a normalization factor.)